Journal article
MULTIPARTITE RATIONAL FUNCTIONS
Documenta mathematica Journal der Deutschen Mathematiker-Vereinigung, v 25, pp 1285-1313
01 Jan 2020
Abstract
Consider a tensor product of free algebras over a field k, the so-called multipartite free algebra A = k<X-(1)> circle times ... circle times k<X-(G)>. It is well-known that A is a domain, but not a fir nor even a Sylvester domain. Inspired by recent advances in free analysis, formal rational expressions over A together with their matrix representations in Mat(n1)(k)circle times...circle times Mat(nG)(k) are employed to construct a skew field of fractions u of A, whose elements are called multipartite rational functions. It is shown that u is the universal skew field of fractions of A in the sense of Cohn. As a consequence a multipartite analog of Amitsur 's theorem on rational identities relating evaluations in matrices over k to evaluations in skew fields is obtained. The characterization of u in terms of matrix evaluations fits naturally into the wider context of free noncommutative function theory, where multipartite rational functions are interpreted as higher order noncommutative rational functions with an associated difference-differential calculus and linear realization theory. Along the way an explicit construction of the universal skew field of fractions of D circle times k for an arbitrary skew field D is given using matrix evaluations and formal rational expressions.
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Details
- Title
- MULTIPARTITE RATIONAL FUNCTIONS
- Creators
- Igor Klep - University of LjubljanaVictor Vinnikov - Ben-Gurion University of the NegevJurij Volcic - Texas A&M Univ, Dept Math, College Stn, TX 77843 USA
- Publication Details
- Documenta mathematica Journal der Deutschen Mathematiker-Vereinigung, v 25, pp 1285-1313
- Publisher
- Fiz Karlsruhe-Leibniz-Inst Informationsinfrastruktur
- Number of pages
- 29
- Grant note
- J18132; J1-2453; P1-0222 / Slovenian Research Agency; Slovenian Research Agency - Slovenia University of Auckland Doctoral Scholarship SCHW 1723/1-1 / Deutsche Forschungsgemeinschaft (DFG); German Research Foundation (DFG) DMS 1954709 / NSF; National Science Foundation (NSF) Marsden Fund Council of the Royal Society of New Zealand; Royal Society of New Zealand; Marsden Fund (NZ)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000592702600040
- Other Identifier
- 991021861881604721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics